L(s) = 1 | + (−0.267 − 0.279i)2-s + (0.0381 − 0.849i)4-s + (−0.104 − 0.994i)7-s + (−0.538 + 0.470i)8-s + (0.447 + 0.894i)9-s + (0.995 + 0.0896i)11-s + (−0.249 + 0.294i)14-s + (−0.571 − 0.0514i)16-s + (0.130 − 0.363i)18-s + (−0.240 − 0.302i)22-s + (0.438 − 1.11i)23-s + (0.999 + 0.0299i)25-s + (−0.849 + 0.0508i)28-s + (0.172 + 0.278i)29-s + (0.584 + 0.732i)32-s + ⋯ |
L(s) = 1 | + (−0.267 − 0.279i)2-s + (0.0381 − 0.849i)4-s + (−0.104 − 0.994i)7-s + (−0.538 + 0.470i)8-s + (0.447 + 0.894i)9-s + (0.995 + 0.0896i)11-s + (−0.249 + 0.294i)14-s + (−0.571 − 0.0514i)16-s + (0.130 − 0.363i)18-s + (−0.240 − 0.302i)22-s + (0.438 − 1.11i)23-s + (0.999 + 0.0299i)25-s + (−0.849 + 0.0508i)28-s + (0.172 + 0.278i)29-s + (0.584 + 0.732i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0640 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0640 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.163346682\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.163346682\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.104 + 0.994i)T \) |
| 11 | \( 1 + (-0.995 - 0.0896i)T \) |
| 43 | \( 1 + (0.998 - 0.0598i)T \) |
good | 2 | \( 1 + (0.267 + 0.279i)T + (-0.0448 + 0.998i)T^{2} \) |
| 3 | \( 1 + (-0.447 - 0.894i)T^{2} \) |
| 5 | \( 1 + (-0.999 - 0.0299i)T^{2} \) |
| 13 | \( 1 + (-0.646 - 0.762i)T^{2} \) |
| 17 | \( 1 + (-0.337 - 0.941i)T^{2} \) |
| 19 | \( 1 + (-0.0149 - 0.999i)T^{2} \) |
| 23 | \( 1 + (-0.438 + 1.11i)T + (-0.733 - 0.680i)T^{2} \) |
| 29 | \( 1 + (-0.172 - 0.278i)T + (-0.447 + 0.894i)T^{2} \) |
| 31 | \( 1 + (0.842 + 0.538i)T^{2} \) |
| 37 | \( 1 + (-0.748 + 1.68i)T + (-0.669 - 0.743i)T^{2} \) |
| 41 | \( 1 + (0.936 + 0.351i)T^{2} \) |
| 47 | \( 1 + (-0.858 + 0.512i)T^{2} \) |
| 53 | \( 1 + (-1.73 - 0.970i)T + (0.525 + 0.850i)T^{2} \) |
| 59 | \( 1 + (-0.134 - 0.990i)T^{2} \) |
| 61 | \( 1 + (0.842 - 0.538i)T^{2} \) |
| 67 | \( 1 + (1.56 + 0.235i)T + (0.955 + 0.294i)T^{2} \) |
| 71 | \( 1 + (0.0888 + 1.48i)T + (-0.992 + 0.119i)T^{2} \) |
| 73 | \( 1 + (-0.971 - 0.237i)T^{2} \) |
| 79 | \( 1 + (0.0248 - 0.116i)T + (-0.913 - 0.406i)T^{2} \) |
| 83 | \( 1 + (0.887 + 0.460i)T^{2} \) |
| 89 | \( 1 + (-0.988 + 0.149i)T^{2} \) |
| 97 | \( 1 + (0.393 + 0.919i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.878256857730604926198422871891, −7.88214461383770375680773340760, −7.01487674497241432886262953152, −6.57148792947111646026452280685, −5.58338479029877687162395607130, −4.67575703308752219030270213941, −4.14365183741559101594158752792, −2.86908742456459571533962099597, −1.79849190215475754223406823090, −0.889239620016045217468450834875,
1.31670373236893675873433089398, 2.72301904097828477783809125094, 3.43545160757465384933291437331, 4.21284206014437866516996378363, 5.25391335322839563041162557081, 6.31820743740654256026453243374, 6.69387894879919205879003367475, 7.46753350541845014253646813828, 8.456276385639504672122274378194, 8.859813899878277202938434797652